## Abstract

The global reforms to public pension schemes over the last thirty years have progressively reduced individuals’ post-retirement social security income. In order to compensate for this, individuals join pension funds and individual plans to increase their wealth at retirement. These types of fully funded plans generally give individuals the opportunity to withdraw the capital accumulated into their scheme or to convert it into an annuity. In this paper, we analyse individuals’ post-retirement choices to allocate the wealth at retirement between consumption, risk-free investments and a life annuity. We develop a discrete time optimisation model, in a deterministic framework, with a constant relative risk aversion (CRRA) utility function. We study the effect of a bequest motive and the annuity rate used by the insurer on the optimal choice. Several numerical applications are presented to illustrate the optimal annuitisation decision results and the optimal consumption paths.

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## Notes

We define the optimised aggregated discounted utility for a fixed \(\alpha \) by \(\xi (\alpha ){=} \max _{C_0,\ldots ,C_{T-1}} \sum _{t{=}0}^{T-1}\left[ U(C_{t})\frac{_{t}p_{x}}{\left( 1+\delta \right) ^t}+\beta U(W_{t+1})\left( 1{-}p_{x+t}\right) \frac{_{t}p_{x}}{\left( 1+\delta \right) ^{t+1}} \right] \).

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## Acknowledgements

The authors would like to thank the anonymous referees. They also acknowledge support by the FNRS cross-university grant PDR *Risk management and Pricing in Finance and Insurance*.

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## Appendix

### Appendix

In this section, we develop the special case without bequest (\(\beta =0\)), still under the positivity condition, in order to motivate the two examples presented in Sect. 3. In this special case, we will obtain an explicit value of the optimal aggregated discounted utility and derive a natural rule to check the level of optimal annuitisation, assuming that the positivity condition is fulfilled and that \(W(T)=0\).

Again, we first consider a fixed percentage \(\alpha \) and determine the optimal consumptions for the optimisation problem with this fixed \(\alpha \). Based on the derivation of the previous section, one obtains the analogue of Eq. (21) with \(\beta =0\):

Recursively substituting \(C_1\), \(C_2\), \(\dots ,C_{T-1} \) into this Eq. (21) (with \(\beta =0\)), the following expression for the optimal initial consumption, \(C_{0}\), can be easily derived:

We notice that this Eq. (25) is, in fact, a special case of Eq. (23) (with \(\beta =0\)). The advantage of the present Eq. (25) is that it offers an explicit expression for \(C_0\). All optimal consumptions, \(C_t\), then follow as a function of the fixed \(\alpha \) from a substitution of \(C_0\) in Eq. (24).

As a next step, we use the fact that we know the optimal consumption for a fixed proportion \(\alpha \) and concentrate now upon the remaining problem, which is an optimisation over \(\alpha \), namely \(\max _{\; 0\le \alpha \le 1} \;\xi \left( \alpha \right) \) with the function \(\xi (\alpha )\), equal to the optimised aggregate discounted utility, defined by:

We first point out that, for \(W_{0}>0\), this function, \(\xi \left( \alpha \right) \), will be defined when \(1-\alpha +\frac{\alpha }{\ddot{a}_x}\sum _{t=0}^{T-1}\left[ e^{R_f}\right] ^{-t}>0,\) which we assume from now on. Further, the sign of the derivative of the function \(\xi \left( \alpha \right) \) is positive when

which will be satisfied if \(\ddot{a}_{x}< \sum _{t=0}^{T-1}[e^{R_f}]^{-t}\) and, according to the Weierstrass theorem, the maximum on the interval [0, 1] equals then \(\alpha =1\). For \(\ddot{a}_{x}>\sum _{t=0}^{T-1}[e^{R_f}]^{-t}\), the derivative is negative, and the maximum is then obtained for \(\alpha =0\). For \(\ddot{a}_{x}=\sum _{t=0}^{T-1}[e^{R_f}]^{-t}\), the function is a constant for each value of \(\alpha \).

Therefore, the final optimisation result, under the positivity condition, for a person with no bequest motive is a zero-one solution. If \(\ddot{a}_{x}>\sum _{t=0}^{T-1}[e^{R_f}]^{-t}\), it will be optimal to invest all the wealth into the financial risk-less asset and and not choose annuitisation (i.e. \(\alpha =0\)). In this case, the annuity seems too expensive. If \(\ddot{a}_{x}<\sum _{t=0}^{T-1}[e^{R_f}]^{-t}\), the optimal solution is to annuitise all the starting endowment by choosing \(\alpha =1\). In this case, the price of a life annuity is less than the price of a certain annuity, which could be considered as a normal situation. Finally, if \(\ddot{a}_{x}=\sum _{t=0}^{T-1}[e^{R_f}]^{-t}\), there is indifference to choosing to annuitise or not.

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Deelstra, G., Devolder, P. & Melis, R. Optimal annuitisation in a deterministic financial environment.
*Decisions Econ Finan* **44**, 161–175 (2021). https://doi.org/10.1007/s10203-020-00316-5

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DOI: https://doi.org/10.1007/s10203-020-00316-5